Inequalities involving $\pi \left(x\right)$

Alzer, Horst; Kwong, Man Kam; Sándor, József

Rendiconti del Seminario Matematico della Università di Padova, Tome 147 (2022), pp. 237-251.

We present several inequalities involving the prime-counting function $\pi \left(x\right)$. Here, we give two examples of our results. We show that

$$\frac{16}{9}\pi \left(x\right)\pi \left(y\right)\le {\pi }^2(x+y)$$

is valid for all integers $x,y\ge 2$. The constant factor $16/9$ is the best possible. The special case $x=y$ leads to

$$\frac{4}{3}\le \frac{\pi \left(2x\right)}{\pi \left(x\right)}(x=2,3,...),$$

where the lower bound $4/3$ is sharp. This complements Landau’s well-known inequality

$$\frac{\pi \left(2x\right)}{\pi \left(x\right)}\le 2(x=2,3,...).$$

Moreover, we prove that the inequality

$${\left(2\frac{\pi (x+y)}{x+y}\right)}^s\le {\left(\frac{\pi \left(x\right)}{x}\right)}^s+{\left(\frac{\pi \left(y\right)}{y}\right)}^s(0<s\in ℝ)$$

holds for all integers $x,y\ge 2$ if and only if $s\le s_0=0.94745....$ Here, $s_0$ is the only positive solution of

$${\left(\frac{16}{7}\right)}^t-{\left(\frac{6}{5}\right)}^t=1.$$

Classification : 11